advertisement

MATH 131-503 Fall 2015 c Wen Liu 5.2 5.2 The Definite Integral Definition: If f is a function defined for a ≤ x ≤ b, we divide the interval [a, b] into n subintervals of equal width b−a 4x = n We let a = x0 , x1 , x2 , · · · , xn−1 , xn = b be the endpoints of these subintervals and we let x∗1 , x∗2 , · · · , x∗n be any sample points in these subintervals, so x∗i lies in the ith subinterval [xi−1 , xi ]. Then the definite integral of f from a to b is Z b n X f (x)dx = lim f (x∗i )4x n→∞ a i=1 provided that this limit exists. If it does exist, we say that f is integrable on [a, b]. Note: R Rb • The symbol is called an integral sign. In the notation a f (x)dx, f (x) is called the integrand and a, b are called the limits of integration; a is the lower limit and b is the upper Rb limit. The dx indicates that the independent variable is x. a f (x)dx is all one symbol. The procedure of calculating an integral is called integration. Rb • The definite integral a f (x)dx is a number; it does not depend on x. Z b b Z Z a a • The sum n X b f (s)ds f (t)dt = f (x)dx = a f (x∗i )4x i=1 that occurs in Definition is called a Riemann sum. • If f takes on both positive and negative values, as in the following figure, then Z b f (x)dx = A1 − A2 a where A1 is the area of the region above the x-axis and below the graph of f , and A2 is the area of the region below the x-axis and above the graph of f . Page 1 of 4 MATH 131-503 Fall 2015 c Wen Liu 5.2 Examples: 1. (p. 346) Consider f (x) = x3 − 6x. Evaluate the Riemann sum for 0 ≤ x ≤ 3 with six subintervals, taking the sample points to be right endpoints. 2. (p. 348) Evaluate R3 0 (x − 1)dx by interpreting it in terms of areas. 3. The graph of g consists of two straight lines and a semicircle. Use it to evaluate R 12 3 g(x)dx. Page 2 of 4 MATH 131-503 Fall 2015 c Wen Liu 5.2 If we choose x∗i to be the midpoints, we get the following approximation. Midpoint Rule: Z b f (x)dx ≈ a n X f (xi )4x = 4x (f (x1 ) + · · · + f (xn )) i=1 where 4x = and xi = b−a n xi−1 + xi = midpoint of [xi−1 , xi ] 2 Examples: Z 4. (p. 349) Use the Midpoint Rule with n = 5 to approximate 1 2 1 dx. x 5. The table gives the values of a function obtained from an experiment. Use them to estimate R9 f (x)dx using three equal subintervals with left endpoints. If the function is known to be an 3 increasing function, can you say whether your estimate is less than or greater than the exact value of the integral? x f (x) 3 4 5 6 7 8 9 -3.4 -2.2 -0.5 0.2 0.9 1.3 1.7 Page 3 of 4 MATH 131-503 Fall 2015 c Wen Liu 5.2 Properties of the Definite Integral Z a • f (x)dx = 0 a Z b a Z f (x)dx = − • f (x)dx b a Z b • Z f (x)dx = a Z c Z f (x)dx + a b f (x)dx c b • cdx = c(b − a), where c is any constant a Z b • Z f (x)dx, where c is any constant cf (x)dx = c a a Z b b Z g(x)dx a a a b Z f (x)dx ± (f (x) ± g(x)) dx = • b Examples: R 10 R8 R 10 6. If it is know that 0 f (x)dx = 17 and 0 f (x)dx = 12, find 8 f (x)dx. 7. Write as a single integral in the form 8. Find R9 0 Rb a f (x)dx for R2 f (x)dx + −5 R5 2 f (x)dx − R −3 −5 f (x)dx. f (x)dx if f (x) = ( 7, if x < 7 x, if x ≥ 7 Page 4 of 4